Sequential Probability Ratio Test (SPRT)

SPRT is provably optimal in the sense of minimizing the average sample number (ASN) required to reach a decision. Unlike fixed-sample-size tests, SPRT processes observations one at a time and terminates as soon as the cumulative evidence crosses an upper or lower boundary.

• Wald's Identity: Guarantees that the test terminates with probability 1.
• Efficiency: Requires on average 50% fewer samples than fixed-sample tests for the same error probabilities.
• Decision Boundaries: Defined by constants A and B derived from the desired false alarm and miss probabilities.

The core mechanism of SPRT is the recursive accumulation of the log-likelihood ratio (LLR). After each observation, the LLR for that sample is computed and added to a running sum.

• Recursive Form: S_n = S_{n-1} + log[ P(x_n | H_1) / P(x_n | H_0) ]
• Threshold Comparison: If S_n >= log(A), accept H_1. If S_n <= log(B), accept H_0. Otherwise, continue sampling.
• Memory Efficiency: Only the current cumulative sum needs to be stored, not the entire observation history.

SPRT provides strict, mathematically guaranteed bounds on the probabilities of Type I (false alarm) and Type II (miss) errors. The thresholds A and B are directly derived from these user-specified error tolerances.

• Wald's Approximations: A ≈ (1 - β) / α and B ≈ β / (1 - α), where α is the false alarm probability and β is the miss probability.
• Conservative Bounds: The actual error probabilities are always less than or equal to the nominal values used to set the thresholds.
• No Asymptotic Assumptions: These guarantees hold for any finite sample size, not just in the limit.

In Automatic Modulation Classification (AMC), SPRT is used to sequentially decide between two candidate modulation schemes (e.g., QPSK vs. 16-QAM) by evaluating the likelihood of each incoming IQ sample.

• Binary Discrimination: SPRT is inherently a binary test; multi-class problems are handled via hierarchical or pairwise SPRT structures.
• Likelihood Models: Requires accurate probabilistic models for the received signal under each modulation hypothesis, often assuming AWGN channels.
• Rapid Decision: Enables cognitive radios to quickly identify the modulation format of an intercepted signal and adapt their receiver configuration.

A critical practical limitation of SPRT is its sensitivity to model mismatch. The optimality guarantees hold only when the assumed likelihood functions perfectly match the true data distribution.

• Nuisance Parameters: Unknown channel parameters (phase offset, frequency offset, SNR) must be estimated or marginalized, leading to composite hypothesis variants like GLRT or ALRT.
• Robustness Trade-off: Sequential tests with mismatched models can exhibit significantly inflated error rates or fail to terminate.
• Mitigation: Hybrid approaches combine SPRT with robust pre-processing or adaptive threshold adjustment.

SPRT is the foundational procedure of Wald's Sequential Analysis, developed by Abraham Wald in the 1940s for wartime quality control. It established the mathematical framework for all sequential decision-making.

• Historical Origin: Developed to reduce sample sizes in destructive testing of munitions.
• Generalization: Extended to multi-hypothesis testing via the Sequential Generalized Likelihood Ratio Test (SGLRT).
• Modern Relevance: Forms the theoretical basis for active learning, bandit algorithms, and anytime inference in modern AI systems.

The foundational theory developed by Abraham Wald in the 1940s that established SPRT. Unlike fixed-sample tests, Wald's framework processes observations one at a time and makes one of three decisions after each sample:

• Accept the null hypothesis
• Accept the alternative hypothesis
• Continue sampling This sequential approach guarantees a decision with fewer observations on average than any fixed-sample test with the same error probabilities, making it optimal for minimizing sample size.

SPRT employs two constant thresholds, A and B, derived from the specified error probabilities:

• Upper threshold A: If the cumulative log-likelihood ratio exceeds this, accept the alternative hypothesis
• Lower threshold B: If it falls below this, accept the null hypothesis
• Between A and B: Continue sampling These boundaries are calculated as functions of the desired Type I error (α) and Type II error (β) probabilities, ensuring the test respects the prescribed error constraints exactly.

Standard SPRT assumes simple hypotheses with fully specified distributions. In modulation classification, unknown parameters like carrier phase, frequency offset, or channel gain create composite hypotheses. Extensions include:

• Weighted SPRT: Averages likelihood over prior distributions of nuisance parameters
• Generalized SPRT: Replaces unknown parameters with maximum likelihood estimates
• Adaptive SPRT: Updates parameter estimates as more samples arrive These variants trade optimality for practical applicability in real-world signal environments.

A practical modification that imposes a maximum sample limit to prevent unbounded test duration. When the truncation point is reached without crossing a boundary, a forced decision is made using:

• The sign of the accumulated log-likelihood ratio
• A secondary threshold optimized for the truncation point Truncation introduces a slight increase in error probabilities but guarantees bounded latency, which is essential for real-time modulation classification systems operating under strict timing constraints in tactical or cognitive radio applications.